This text is an entry in the MAA Guides series, each volume of which is intended to provide a short, concentrated summary of a primary mathematical subject. This volume on real analysis focuses on the material in a standard first graduate course. It is ideally suited to serve as a quick look for someone new to the subject, for review, or as preparation for qualifying exams. It is concise, very clearly written, and full of little nuggets of insight.
The author has written a full scale graduate real analysis book of his own, Real Analysis: Modern Techniques and Their Applications and the current book owes much to the structure of the real analysis course that the author envisions there. The contents also map pretty directly to the real analysis portion of Rudin’s Real and Complex Analysis and to Royden’s Real Analysis. Anyone who has taken such a course will have seen all the topics, with the possible exception of the last section (on distributions, aka generalized functions). It is assumed that the reader is familiar with classical real variables at the level, for example, of Rudin’s Principles of Mathematical Analysis.
The book contains essential definitions, major theorems, and some carefully chosen examples. Sketches of proofs for theorems are often included but technical details are not. The author manages to accomplish quite a bit in just a few pages. For example, with just three simple examples and a brief summary, the author sorts out the convergence relationships (pointwise, uniform, almost everywhere, in L1. in measure) for sequences of functions. Although both concise and rigorous, the author manages to be easier – and more pleasant – to read than Rudin, partly because he’s not at all reluctant to put in the “and this means…” or “the idea of the proof is…” parts that Rudin routinely leaves out.
Although this would not be my book of choice for learning graduate level real analysis, I would highly recommend it to those studying for qualifying exams, to anyone looking for a quick review, or as a supplement to a standard textbook.
Bill Satzer (firstname.lastname@example.org) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.